Model reduction of time-delay systems using position balancing and delay Lyapunov equations

  • Elias Jarlebring
  • Tobias Damm
  • Wim Michiels
Original Article


Balanced truncation is a standard and very natural approach to approximate dynamical systems. We present a version of balanced truncation for model order reduction of linear time-delay systems. The procedure is based on a coordinate transformation of the position and preserves the delay structure of the system. We therefore call it (structure-preserving) position balancing. To every position, we associate quantities representing energies for the controllability and observability of the position. We show that these energies can be expressed explicitly in terms of the solutions to corresponding delay Lyapunov equations. Apart from characterizing the energies, we show that one block of the (operator) controllability and observability Gramians in the operator formulation of the time-delay system can also be characterized with the delay Lyapunov equation. The delay Lyapunov equation undergoes a contragredient transformation when we apply the position coordinate transformation and we propose to truncate it in a classical fashion, such that positions which are only weakly connected to the input and the output in the sense of the energy concepts are removed.


Time-delay systems Model reduction Balanced truncation  Lyapunov equations  



This work has been supported by the Programme of Interuniversity Attraction Poles of the Belgian Federal Science Policy Office (IAP P6-DYSCO), by OPTEC, the Optimization in Engineering Center of the K.U. Leuven, and by the project STRT1-09/33 of the K.U. Leuven Research Council. The first author is supported by the Dahlquist research fellowship.


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Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  1. 1.Department of MathematicsKTHStockholmSweden
  2. 2.Technische Universität Kaiserslautern, Fachbereich Mathematik, Erwin Schrödinger StraßeKaiserslauternGermany
  3. 3.Department of Computer ScienceKU LeuvenHeverleeBelgium

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